Odd that it's saying it's being updated to 0.621, as the current version is 0.63 (I checked all the servers, and it's up properly).
It sounds like, for whatever reason, the previous version (0.621) is being cached by your computer, but it's still downloading the correct version 0.63 (hence the constant notifications).
I know this is a lame recommendation, but have you tried rebooting?